If $a : b = 3 : 4$, $b : c = 4 : 7$, then $\frac{a + b + c}{c}$ is equal to (Hotel Management, 2010)

### Concept & Chaining Ratios When linking two separate ratios that share a common term (in this case, $b$), check if the value of that common term is identical in both ratios. If it is, you can directly combine them into a single continuous ratio $a : b : c$. ### Step-by-Step Solution - **Given Ratios:** $a : b = 3 : 4$ and $b : c = 4 : 7$. - **Combine Ratios:** Since $b$ is represented by $4$ in both ratios, we can directly link them. $a : b : c = 3 : 4 : 7$. - **Assign Proportional Values:** Let $a = 3$, $b = 4$, and $c = 7$. (Because the target expression is a ratio itself, the constant multiplier $k$ will cancel out). - **Evaluate Expression:** - Target: $\frac{a + b + c}{c}$ - Substitute: $\frac{3 + 4 + 7}{7}$ - Simplify: $\frac{14}{7} = 2$. ### Exam Strategy & Shortcut Notice that $b=4$ in both given conditions. Instantly write down $a=3, b=4, c=7$. Add them up in your head to get $14$, and divide by $c$ ($7$). The answer is $2$. This should be solvable mentally in under 5 seconds. ### Common Pitfall Overcomplicating the problem by introducing variables like $a=3k, b=4k$ is unnecessary when evaluating a homogeneous fractional expression, as it just adds extra writing. ### Final Answer Therefore, the correct answer is **$2$**.